The only place I know mewtwo is available at is the cerulean city cave (for those who don't know,you have to beat the elite four to have access to it).I always used my masterball on mewtwo and ultra balls on the 3 birds...but here is how..
Step 1 - have you party all level +70 & teach some of your Pokemon status moves (i recommend hypnosis,sing or sleep powder)
Step 2 - buy some ultra balls (cinnabar),about 2-3 max revives,5 full restores/super or hyper potions & 5 max elixers (Pokemon league)
Step 3 - make your way through the cave to mewtwo.
Step 4 - once you reach mewtwo,save your game in case mewtwo starts giving you a tough time or if you run out of items.
Step 5 - start the battle & do your best to weaken mewtwo without making him faint (if he faints,reset your game and try again)
Step 6 - once mewtwo is in the red,try putting him to sleep & use your ultra balls (if mewtwo is asleep,you should be able to capture him)
Hope this helped :)
You can catch Mewtwo easily by using a Master Ball on it or by using an Ultra Ball on it after lowering its HP.
since you're in a cave,i could recommend a dusk ball.
I prefer to use my Master Ball for Mewtwo, but at least use an Ultra Ball.
Master Ball
You can probaly catch him with a ultra ball but I would use the master ball
master ball ultra ball quick ball or a premier ball
With a master ball.
Throw a Master Ball at it.
Well it matters what game you have, if you have Pokemon fire red, you can catch it with a master ball
to catch suicune,mewtwo,raikou and entei
The only sure way to instantly catch any Pokemon is with a Master Ball.
a master ball preferably but any ball will work with a bit of luck